Unfortunately, the MILCs haven't been around long enough to have good old bodies yet, so finding one with decent features for cheap is going to be tough, and $300 is not much to spend: that's basically a new P&S.

I would seriously look at something like an RX100... small enough to have with you, but sophisticated (and fast) enough to take actual photographs with DOF control, natural light, etc. It's a bit pricier, but a fantastic learning tool (and a great second camera to back up a future ILC).

-- hide signature --

If you take a great photograph, I really don't care whether it came from a Leica or smartphone. Just share it.